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Properties

Label 2-72450-1.1-c1-0-84
Degree $2$
Conductor $72450$
Sign $-1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s + 2·11-s + 2·13-s − 14-s + 16-s + 2·17-s − 4·19-s − 2·22-s − 23-s − 2·26-s + 28-s − 2·29-s − 8·31-s − 32-s − 2·34-s + 4·37-s + 4·38-s + 12·41-s + 2·43-s + 2·44-s + 46-s − 8·47-s + 49-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.426·22-s − 0.208·23-s − 0.392·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.657·37-s + 0.648·38-s + 1.87·41-s + 0.304·43-s + 0.301·44-s + 0.147·46-s − 1.16·47-s + 1/7·49-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.38418827839482, −14.16035218099386, −13.10214568742725, −12.88373361106140, −12.38960141426471, −11.55337737951594, −11.38603519857441, −10.79734115986760, −10.41907477055365, −9.629661798452999, −9.330817167123573, −8.765303686876694, −8.303953069119608, −7.645232142854502, −7.388350225606726, −6.566734716352575, −6.089216632619963, −5.697896939014924, −4.844109577694963, −4.226144388230616, −3.656652578922234, −2.999085091900453, −2.154164773663200, −1.625080577992708, −0.9427708918793290, 0, 0.9427708918793290, 1.625080577992708, 2.154164773663200, 2.999085091900453, 3.656652578922234, 4.226144388230616, 4.844109577694963, 5.697896939014924, 6.089216632619963, 6.566734716352575, 7.388350225606726, 7.645232142854502, 8.303953069119608, 8.765303686876694, 9.330817167123573, 9.629661798452999, 10.41907477055365, 10.79734115986760, 11.38603519857441, 11.55337737951594, 12.38960141426471, 12.88373361106140, 13.10214568742725, 14.16035218099386, 14.38418827839482

Graph of the $Z$-function along the critical line